A280089 -id:A280089 - cp's OEIS frontend

A033430 a(n) = 4*n^3.

0, 4, 32, 108, 256, 500, 864, 1372, 2048, 2916, 4000, 5324, 6912, 8788, 10976, 13500, 16384, 19652, 23328, 27436, 32000, 37044, 42592, 48668, 55296, 62500, 70304, 78732, 87808, 97556, 108000, 119164, 131072, 143748, 157216, 171500, 186624, 202612, 219488

Offset: 0

Jeff Burch

2*a(n) = (2*n)^3 is a perfect cube.
Number of edges of the product of two complete bipartite graphs, each of order 2n, K_n,n x K_n,n - Roberto E. Martinez II, Jan 07 2002
This sequence is related to A049451 by a(n) = n*A049451(n) + sum( A049451(i), i=0..n-1 ) for n>0. - Bruno Berselli, Dec 19 2013
For n>=3, also the detour index of the n-gear graph. - Eric W. Weisstein, Dec 20 2017
For n > 0, this sequence can be obtained by summing consecutive blocks of odd numbers where the n-th block contains the next 2n odd numbers. - Marco Zárate, Jun 15 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..750
Frank Ellermann, Illustration of binomial transforms
Eric Weisstein's World of Mathematics, Detour Index
Eric Weisstein's World of Mathematics, Gear Graph
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A049451.
Cf. A000292, A007588.
Cf. A001844.

[4*n^3: n in [0..40]]; // Vincenzo Librandi, Apr 27 2011

4 Range[0, 40]^3 (* Harvey P. Dale, Sep 07 2016 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 4, 32, 108}, 40] (* Harvey P. Dale, Sep 07 2016 *)
Table[4 n^3, {n, 0, 20}] (* Eric W. Weisstein, Dec 20 2017 *)
CoefficientList[Series[(4 x (1 + 4 x + x^2))/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 20 2017 *)

a(n)=4*n^3 \\ Charles R Greathouse IV, Oct 07 2015

G.f. 4*x*(1+4*x+x^2)/ (x-1)^4. - R. J. Mathar, Feb 01 2011
From Ilya Gutkovskiy, May 25 2016: (Start)
E.g.f.: 4*x*(1 + 3*x + x^2)*exp(x).
Sum_{n>=1} 1/a(n) = zeta(3)/4. (End)
Product_{n>=1} a(n)/A280089(n) = Pi. - Daniel Suteu, Dec 26 2016
From Bruce J. Nicholson, Dec 07 2019: (Start)
a(n) = 24*A000292(n-1) + 4*n.
a(n) = 2*A007588(n) + 2*n. (End)
a(n) = Sum_{k=0..2*n-1} (2*n*(n-1)-2*k+1). - Sean A. Irvine, Jun 19 2025

A298637 Triangular array of a Catalan number variety: T(n,k) is the number of words consisting of n parentheses containing k well-balanced pairs.

Original entry on oeis.org

1, 2, 3, 1, 4, 4, 5, 9, 2, 6, 16, 10, 7, 25, 27, 5, 8, 36, 56, 28, 9, 49, 100, 84, 14, 10, 64, 162, 192, 84, 11, 81, 245, 375, 270, 42, 12, 100, 352, 660, 660, 264, 13, 121, 486, 1078, 1375, 891, 132, 14, 144, 650, 1664, 2574, 2288, 858, 15, 169, 847, 2457, 4459, 5005, 3003, 429

Offset: 0

Marko Riedel, Jan 23 2018

A well-balanced run in a word of parentheses is a maximal run where every initial segment of the run has at least as many left parentheses as right ones and the number of open parentheses is the same as that of closed ones. The variable k in the sequence definition is the sum of the count of balanced pairs in all maximal runs in the word and n is the length of the word. Runs are maximal substrings counted by ordinary Catalan numbers.

			The word ))))(()(()))((() contains five well-balanced pairs of parentheses.
Triangular array T(n,k) begins:
   1;
   2;
   3,   1;
   4,   4;
   5,   9,   2;
   6,  16,  10;
   7,  25,  27,   5;
   8,  36,  56,  28;
   9,  49, 100,  84,  14;
  10,  64, 162, 192,  84;
  11,  81, 245, 375, 270,  42;
  12, 100, 352, 660, 660, 264;

Alois P. Heinz, Rows n = 0..200, flattened
Toufik Mansour, Armend Sh. Shabani, Bargraphs in bargraphs, Turkish Journal of Mathematics (2018) Vol. 42, Issue 5, 2763-2773.
Marko Riedel et al., Generalisation for Catalan number.
Marko Riedel, Maple code for A298637 including enumeration, generating function, and two closed forms.

Row sums give A000079.
T(2n,n) gives A000108.
T(2n+1,n) gives A068875. T(n,1) gives A000290. T(2n,2) gives A280089.
Cf. A007318, A061554.

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i>0, x, 1)*b(n-1, max(0, i-1))+b(n-1, i+1)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..16);  # Alois P. Heinz, Jan 23 2018

Table[((n + 1 - 2 k)^2/(n + 1)) Binomial[n + 1, k], {n, 0, 17}, {k, 0, Floor[n/2]}] // Flatten (* Michael De Vlieger, Jan 23 2018 *)

T(n,k) = ((n+1-2*k)^2/(n+1))*C(n+1,k) where 0 <= k <= floor(n/2).
Bivariate o.g.f. is C(u*z^2)/(1-z*C(u*z^2))^2 with u counting pairs of parentheses and z counting total word length where C(z) = (1-sqrt(1-4*z))/(2*z) is the o.g.f. of the Catalan numbers.
T(2*k,k) = C(k), the k-th Catalan number.
T(n,0) = n+1 by construction.

cp's OEIS Frontend

A033430 a(n) = 4*n^3.

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